They were first discovered
in 1939 by Arthur Stone, who set up a Flexagon Committee to investigate their
properties.

The committee consisted
of the famous Mathematicians and Physicists, B. Tuckerman, R.P.
Feynman and J.W. Tukey.

Trihexaflexagons

The trihexaflexagon has
three different faces. To make a trihexaflexagon follow these easy steps.

Step
1:

Cut a
strip of paper and mark ten equilateral triangles as shown on the left.
Number the triangles and crease along the lines joining the triangles.
Fold the strip back along the line marked ab.

Step 2:

Fold
the strip back along the line marked cd.

Step
3:

Fold
the triangle labelled 1 to the back and glue it to the back of the triangle
labelled 3*.

Step
4:

To change
the face of your trihexaflexagon, place your thumb and forefinger on the
two triangles labelled 2* and pinch them together so their backs are touching.
From the centre ease out the inner edge of the two unmarked triangles
so that your trihexaflexagon is flat.

Now you can have some
fun. Colour the three different faces of your trihexaflexagon with different
designs and start flexing. You should find that there are a few more designs
than you counted on!

Hexahexaflexagons

A hexahexaflexagon has
six different faces.

Construct a hexahexaflexagon
as follows:

Step
1:

Take
a longer piece of paper (almost twice as long as the trihexaflexagon)
and mark 19 equilateral triangles as shown. Carefully label the triangles
on both sides of the paper (as shown above) and crease along the lines
joining the triangles.

Step
2:

Fold
so that the triangles labelled 4 face each other, as do the triangles
labelled 5 and the triangles labelled 6. You are essentially wrapping
the paper into a spiral. Your strip should now resemble that used to make
the trihexaflexagon.

Step
3:

Fold
and glue as in the instructions for the trihexaflexagon. You should find
that all the triangles on each side have the same number.

Start flexing and see
if you can find all the other sides. (Some sides are trickier to find than
others.) Colour or paste pictures on the six different faces and flex away.
How many different patterns can you make?